Discrete-Time Market Model

Chapter 2 Discrete-Time Market Model The single-step model considered in Chapter1 is extended to a discrete-time model with N + 1 time instants t = 0, 1, ... , N. A basic limitation of the one-step model is that it does not allow for trading until the end of the first time period is reached, while the multistep model allows for multiple portfolio re-allocations over time. The Cox-Ross-Rubinstein (CRR) model, or binomial model, is considered as an example whose importance also lies with its computer implementability. 2.1 Discrete-Time Compounding . 51 2.2 Arbitrage and Self-Financing Portfolios . 54 2.3 Contingent Claims . 60 2.4 Martingales and Conditional Expectations . 65 2.5 Market Completeness and Risk-Neutral Measures 71 2.6 The Cox-Ross-Rubinstein (CRR) Market Model . 73 Exercises . 78 2.1 Discrete-Time Compounding Investment plan We invest an amount m each year in an investment plan that carries a constant interest rate r. At the end of the N-th year, the value of the amount m invested at the beginning of year k = 1, 2, ... , N has turned into (1 + r)N−k+1m and the value of the plan at the end of the N-th year becomes N X N−k+1 AN : = m (1 + r) (2.1) k=1 51 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault N X = m (1 + r)k k=1 (1 + r)N − 1 = m(1 + r) , r hence the duration N of the plan satisfies 1 rAN N + 1 = log 1 + r + . log(1 + r) m Loan repayment At time t = 0 one borrows an amount A1 := A over a period of N years at the constant interest rate r per year. Proposition 2.1. Constant repayment. Assuming that the loan is completely repaid at the beginning of year N + 1, the amount m refunded every year is given by r(1 + r)N A r m = = A. (2.2) (1 + r)N − 1 1 − (1 + r)−N Proof. Denoting by Ak the amount owed by the borrower at the beginning o of year n k = 1, 2, ... , N with A1 = A, the amount m refunded at the end of the first year can be decomposed as m = rA1 + (m − rA1), into rA1 paid in interest and m − rA1 in principal repayment, i.e. there remains A2 = A1 − (m − rA1) = (1 + r)A1 − m, to be refunded. Similarly, the amount m refunded at the end of the second year can be decomposed as m = rA2 + (m − rA2), into rA2 paid in interest and m − rA2 in principal repayment, i.e. there remains A3 = A2 − (m − rA2) = (1 + r)A2 − m 52 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance = (1 + r)((1 + r)A1 − m) − m 2 = (1 + r) A1 − m − (1 + r)m to be refunded. After repeating the argument we find that at the beginning of year k there remains k−1 k−2 Ak = (1 + r) A1 − m − (1 + r)m − · · · − (1 + r) m k−2 k−1 X i = (1 + r) A1 − m (1 + r) i=0 1 − (1 + r)k−1 = (1 + r)k−1A + m 1 r to be refunded, i.e. m − (1 + r)k−1(m − rA) A = , k = 1, 2, ... , N. (2.3) k r We also note that the repayment at the end of year k can be decomposed as m = rAk + (m − rAk), with k−1 rAk = m + (1 + r) (rA1 − m) in interest repayment, and k−1 m − rAk = (1 + r) (m − rA1) in principal repayment. At the beginning of year N + 1, the loan should be completely repaid, hence AN+1 = 0, which reads 1 − (1 + r)N (1 + r)N A + m = 0, r and yields (2.2). We also have A 1 − (1 + r)−N = . (2.4) m r and 1 m log(1 − rA/m) N = log = − . log(1 + r) m − rA log(1 + r) 53 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault Remark: One needs m > rA in order for N to be finite. The next proposition is a direct consequence of (2.2) and (2.3). Proposition 2.2. The k-th interest repayment can be written as N−k+1 1 X −l rAk = m 1 − = mr (1 + r) , (1 + r)N−k+1 l=1 and the k-th principal repayment is m m − rAk = , k = 1, 2, ... , N. (1 + r)N−k+1 Note that the sum of discounted payments at the rate r is N X m 1 − (1 + r)−N = m = A. (1 + r)l r l=1 In particular, the first interest repayment satisfies N X 1 −N rA = rA1 = mr = m 1 − (1 + r) , (1 + r)l l=1 and the first principal repayment is m m − rA = . (1 + r)N 2.2 Arbitrage and Self-Financing Portfolios Stochastic processes A stochastic process on a probability space (W, F, P) is a family (Xt)t∈T of random variables Xt : W −→ R indexed by a set T . Examples include: • the one-step (or two-instant) model: T = {0, 1}, • the discrete-time model with finite horizon: T = {0, 1, ... , N}, • the discrete-time model with infinite horizon: T = N, • the continuous-time model: T = R+. 54 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance For real-world examples of stochastic processes one can mention: • the time evolution of a risky asset, e.g. Xt represents the price of the asset at time t ∈ T . • the time evolution of a physical parameter - for example, Xt represents a temperature observed at time t ∈ T . In this chapter, we focus on the finite horizon discrete-time model with T = {0, 1, ... , N}. 0 1 2 3 N Asset price modeling The prices at time t = 0 of d + 1 assets numbered 0, 1, ... , d are denoted by the random vector (0) (1) (d) S0 = S0 , S0 , ... , S0 in Rd+1. Similarly, the values at time t = 1, 2, ... , N of assets no 0, 1, ... , d are denoted by the random vector (0) (1) (d) St = St , St , ... , St S on W, which forms a stochastic process t t=0,1,...,N . In the sequel we assume that asset no 0 is a riskless asset (of savings account type) yielding an interest rate r, i.e. we have (0) t (0) St = (1 + r) S0 , t = 0, 1, ... , N. Portfolio strategies Definition 2.3. A portfolio strategy is a stochastic process ξ ⊂ t t=1,2,...,N d+1 (k) o R where ξt denotes the (possibly fractional) quantity of asset n k held in the portfolio over the time interval (t − 1, t], t = 1, 2, ... , N. Note that the portfolio allocation (0) (1) (d) ξt = ξt , ξt , ... , ξt is decided at time t − 1 and remains constant over the interval (t − 1, t] while (k) (k) the stock price changes from St−1 to St over this time interval. 55 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html N. Privault ξ1 ξ2 ξ3 ξN 0 1 2 3 N In other words, the quantity (k) (k) ξt St−1 represents the amount invested in asset no k at the beginning of the time interval (t − 1, t], and (k) (k) ξt St represents the value of this investment at the end of the time interval (t − 1, t], t = 1, 2, ... , N. Self-financing portfolio strategies The opening price of the portfolio at the beginning of the time interval (t − 1, t] is d X (k) (k) ξt • St−1 = ξt St−1, k=0 when the market “opens” at time t − 1. When the market “closes”at the end of the time interval (t − 1, t], it takes the closing value d X (k) (k) ξt • St = ξt St , (2.5) k=0 t = 1, 2, ... , N. After the new portfolio allocation ξt+1 is designed we get the new portfolio opening price d X (k) (k) • ξt+1 St = ξt+1St , (2.6) k=0 at the beginning of the next trading session (t, t + 1], t = 0, 1, ... , N − 1. Note that here, the stock price St is assumed to remain constant “overnight”, i.e. from the end of (t − 1, t] to the beginning of (t, t + 1], t = 1, 2, ... , N − 1. In case (2.5) coincides with (2.6) for t = 0, 1, ... , N − 1 we say that the ξ self-financing portfolio strategy t t=1,2,...,N is . A non self-financing portfolio could be either bleeding money, or burning cash, for no good reason. Definition 2.4. A portfolio strategy ξ is said to be t t=1,2,...,N self-financing if 56 " This version: July 4, 2021 https://personal.ntu.edu.sg/nprivault/indext.html Notes on Stochastic Finance ξt • St = ξt+1 • St, t = 1, 2, ... , N − 1, (2.7) i.e. d d X (k) (k) X (k) (k) ξt St = ξt+1St , t = 1, 2, ... , N − 1. k=0 k=0 | {z } | {z } Closing value Opening price The meaning of the self-financing condition (2.7) is simply that one cannot take any money in or out of the portfolio during the “overnight” transition period at time t. In other words, at the beginning of the new trading session (t, t + 1] one should re-invest the totality of the portfolio value obtained at the end of the interval (t − 1, t].

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